Explanation Given: The differential equation is d y d x = y ln ( y + 2 ) . Calculation: Consider the given differential equation, d y d x = y ln ( y + 2 ) (1) Write the equation of equilibrium and solve for variable y , Substitute 0 for d y d x in equation (1) and solve to find the critical points, y ln ( y + 2 ) = 0 Further solve for factor y = 0 , y = 0 And for factor ln ( y + 2 ) = 0 , ln ( y + 2 ) = 0 y + 2 = 1 y = 1 − 2 y = − 1 Therefore, the critical points are − 1 and 0 . Take the critical points as − 1 and 0 on a vertical line and then defined it into three intervals, as follows, − ∞ < y < − 1 , − 1 < y < 0 and 0 < y < ∞ . Write the differential equation in terms of f ( y ) as follows, f ( y ) = y ln ( y + 2 ) (2) To find the algebraic sign take the points on the constructed intervals and substitute it in equation (2) as below, For interval − ∞ < y < − 1 substitute − 1.5 for y in equation (2), f ( − 1.5 ) = ( − 1.5 ) ln ( − 1.5 + 2 ) = ( − 1.5 ) ln ( 0.5 ) = 1.0397 Therefore for interval − ∞ < y < − 1 the algebraic sign is positive. For interval − 1 < y < 0 substitute − 0.5 for y in equation (2), f ( − 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 2.1, Problem 27E

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To sketch: the typical solution curve in the region in the xy-plane determined by the graph of equilibrium solutions.

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Chapter 2.1, Problem 26E

Chapter 2.1, Problem 28E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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the typical solution curve in the region in the x y -plane determined by the graph of equilibrium solutions.

Solution Summary: The author illustrates the typical solution curve in the xy -plane determined by the graph of equilibrium solutions.

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To sketch: the typical solution curve in the region in the xy-plane determined by the graph of equilibrium solutions.

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Chapter 2 Solutions